Integration without u-substitution

[quiet terrace]

Can someone explain to me why the first image is wrong and the second is correct?
And also what I need to have in mind to to make the correct choices?

[soft orbit]

so

there nothing wrong

its the same

$\frac{1}{3} log({e^{x} - \frac{1}{3}}
) +c= \frac{1}{3} log{\frac{3e^{x}-1}{3}} +c =\frac{1}{3} log({3e^{x}-1}) - \frac{1}{3} log({3}) +c$

[tame lotus]

ᴰᴱᴬᴰ ★ ᴋɪʟʟᴇʀ ⁰⁴

[soft orbit]

and and -1/3 log 3 + c = c

its also a constant

so its the same